Gov. Paterson’s top budget aide warned yesterday that New York’s cash crisis is so dire that the treasury could soon be as much as $1 billion in the red despite a series of radical cost-saving actions.

Budget Director Robert Megna predicted that the “unprecedented” cash crunch would strike sometime in the first week of June, when the state — its coffers depleted from doling out income-tax returns — comes on the hook for up to $10 billion in mandatory payments.

The crush of expenses would devour what then remains of the state’s now $3.4 billion “rainy day” fund and would leave it $1 billion in the hole, Megna told reporters.